Near-quadratic sum-product estimates in fixed number fields
Abstract
Let K be a fixed number field. We prove a quantitative expansion theorem for mixed sumsets A+B, assuming only that A is covered by relatively few cosets of a multiplicative group of small finite valuation rank. The constants are independent of the generators, their heights, and their prime ideal supports. As a consequence, if A⊂ K×, |A|=N≥ 3, and each element has nonzero valuation at at most k finite places, then every B⊂ K with |B|=N satisfies max{|A+B|,|AA|} ≥ N2exp[−CK((k+1)log(log N+2)+√((k+1)log N))]. This is N2−o(1) for k=o(log N/loglog N). We also obtain a trade-off between the additive and multiplicative deficits, estimates for iterated linear sumsets, and a transfer theorem for product constructions A=PG in terms of the multiplicative deficit of P. The proof recovers local scales from an exact sum and bounds the remaining conditional entropy by the overlap of entropy measures on fixed trees.
Related
- An additive problem in connection with some quadratic real fields with class number one
- Ideal class groups of certain real quadratic number fields and factorizations of values of certain quadratic polynomials
- Fermat curves and explicit class field theory for real quadratic number fields
- Finiteness theorems for trace Euclidean quadratic lattices over totally real number fields
- Class numbers and finiteness of norm Euclidean quadratic lattices over algebraic number fields
BibTeX
@misc{zhang2026nearquadratic,
title = {{Near-quadratic sum-product estimates in fixed number fields}},
author = {Kaiqiang Zhang and Yuanguo Zeng and Yuyu Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36930},
eprint = {2610.36930},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36930},
}
RIS
TY - UNPB
TI - Near-quadratic sum-product estimates in fixed number fields
AU - Kaiqiang Zhang
AU - Yuanguo Zeng
AU - Yuyu Wang
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.36930
UR - https://arxiv.org/abs/2610.36930
AB - Let K be a fixed number field. We prove a quantitative expansion theorem for mixed sumsets A+B, assuming only that A is covered by relatively few cosets of a multiplicative group of small finite valuation rank. The constants are independent of the generators, their heights, and their prime ideal supports. As a consequence, if A⊂ K^×, |A|=N≥ 3, and each element has nonzero valuation at at most k finite places, then every B⊂ K with |B|=N satisfies max{|A+B|,|AA|} ≥ N^2exp[−C_K((k+1)log(log N+2)+√((k+1)log N))]. This is N^2−o(1) for k=o(log N/loglog N). We also obtain a trade-off between the additive and multiplicative deficits, estimates for iterated linear sumsets, and a transfer theorem for product constructions A=PG in terms of the multiplicative deficit of P. The proof recovers local scales from an exact sum and bounds the remaining conditional entropy by the overlap of entropy measures on fixed trees.
ER -